A Rigidity Result for Extensions of Braided Tensor C–Categories Derived from Compact Matrix Quantum Groups

• Published in: Communications in mathematical physics Vol. 306; no. 3; pp. 647 - 662
• Main Authors: Pinzari, Claudia, Roberts, John E.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.09.2011; Springer
• Subjects: Algebra; Classical and Quantum Gravitation; Complex Systems; Exact sciences and technology; Group theory; Group theory and generalizations; Mathematical and Computational Physics; Mathematical methods in physics; Mathematical Physics; Mathematics; Nonassociative rings and algebras; Other topics in mathematical methods in physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Sciences and techniques of general use; Theoretical; Topological groups, lie groups; Quantum Group; Fusion Category; Fusion Rule; Braid Group; Tensor Category; Matrix group; Category; Mathematical physics; Classical group
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-011-1260-7

Let G be a classical compact Lie group and G μ the associated compact matrix quantum group deformed by a positive parameter  μ (or in the type A case). It is well known that the category of unitary representations of G μ is a braided tensor C *–category. We show that any braided tensor *–functor to another braided tensor C *–category with irreducible tensor unit is full if | μ | ≠ 1. In particular, the functor of restriction Rep G μ → Rep( K ) to a proper compact quantum subgroup K cannot be made into a braided functor. Our result also shows that the Temperley–Lieb category for d  > 2 can not be embedded properly into a larger category with the same objects as a braided tensor C *–subcategory.